Periodic points in random substitution subshifts

TL;DR

This paper investigates the existence and enumeration of periodic points in random substitution subshifts, introducing new properties and criteria to analyze their structure and periodic behavior.

Contribution

It introduces the disjoint images condition for random substitutions and provides criteria and methods for identifying and counting periodic points.

Findings

Disjoint images condition helps determine periodic points.

Primitive compatible substitutions can admit periodic points.

Method for enumerating periodic points of specified length.

Abstract

We study various aspects of periodic points for random substitution subshifts. In order to do so, we introduce a new property for random substitutions called the disjoint images condition. We provide a procedure for determining the property for compatible random substitutions-random substitutions for which a well-defined abelianisation exists. We find some simple necessary criteria for primitive, compatible random substitutions to admit periodic points in their subshifts. In the case that the random substitution further has disjoint images and is of constant length, we provide a stronger criterion. A method is outlined for enumerating periodic points of any specified length in a random substitution subshift.